...the new tax could impact New Zealand's competitiveness as a tourist destination, especially for those travellers who are price sensitive.To that I will add that it will be more significant for New Zealander travellers than for international visitors to New Zealand. Why? New Zealander travellers have no option when undertaking international travel than to pay the new levy - they will pay the $22 as part of the ticket price travelling to and from New Zealand because New Zealand is the origin of their travel. They can't avoid paying it. International visitors on the other hand have other options - they don't need to travel to New Zealand. Maybe the $22 extra cost is enough to sway them in favour of some other destination.
In terms of relative cost, the levy is probably much less than one percent of the cost of long haul travel to New Zealand so probably won't make a whole lot of difference either way for many long haul travellers. However, the addition of $22 onto a budget fare to/from Australia or the Pacific Islands is quite substantial - around five percent of the cheapest fares on offer. New Zealanders make up a larger proportion of short-haul passengers than long-haul passengers on flights to/from New Zealand (compare departures of New Zealanders with arrivals of international visitors in Statistics New Zealand data here). So, the levy is likely to be more significant for New Zealanders than for international visitors because it affects the cost of flights New Zealanders take relatively more than for international visitors, and New Zealanders have no alternative but to pay the levy. So, New Zealanders should be less price sensitive (because of having no options), but also pay a higher proportionate increase in price for fares (because of the higher proportion of low-cost short-haul fares).
However, coming back to the effects of the tax on price sensitive travellers versus those who are less price sensitive, we can illustrate the difference on a diagram. Assuming a starting price that is the same (P0) [*] and the same starting quantity (Q0), demand by price sensitive travellers is D1 and demand by less-price-sensitive travellers is D2 (note that D1 is flatter than D2). We illustrate the bio-security and customs levy using the S + tax curve (or maybe that should be S + levy?), which is the supply curve shifted upwards by the per-unit value of the levy ($22).
For price sensitive travellers, the price paid by these travellers increases to PC1, the effective price for the airlines falls to PP1 (this is the price the consumers pay, minus the levy), and quantity of tickets falls to QT1. In contrast, for less-price-sensitive travellers, the price paid by these travellers increases to PC2, the effective price for the airlines falls to PP2 (this is the price the consumers pay, minus the levy), and quantity of tickets falls to QT2. Note that the price difference is larger for the less-price-sensitive travellers than price sensitive travellers, and the change in quantity is larger for the price sensitive travellers. So, price sensitive travellers are likely to be turned off travel to/from New Zealand by the levy in greater numbers than less-price-sensitive travellers (as noted by Auckland Airport). For those of you keeping score, New Zealanders face a double-whammy here - higher proportionate increase in fares firstly (as noted above), plus higher fare increases on top of that because of being less price sensitive than other travellers.
Finally, from the diagram the deadweight loss of the levy will be relatively larger from price sensitive travellers (the area BEF is the deadweight loss for price sensitive travellers, compared with the area AED for less-price-sensitive travellers).
One final point: Given the levy will affect travel to/from Australia disproportionately more than travel to/from other destinations, maybe this is a modest way of ensuring a reduction in net migration from Australia, while keeping those who are already here?
[*] This of course would not be true. Airlines are very effective at price discrimination, so less price sensitive travellers typically pay a higher fare than less price sensitive travellers. To see why, think about the difference between paying for a flight long in advance, versus paying a few days before the flight. Travellers paying long in advance typically have many other options for destination, route, date of travel, etc. so are more price sensitive, and pay a lower price. Travellers buying a flight a few days beforehand typically have to be somewhere in a hurry - there are few other options to them, so they are less price sensitive, and pay a higher price. However, for the purposes of this exercise let's say we deflate the price differential so that both prices are on different scales.